Problem: Simplify; express your answer in exponential form. Assume $a\neq 0, z\neq 0$. $\dfrac{{(a^{5}z^{-4})^{2}}}{{(a^{-5}z^{2})^{-1}}}$
To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(a^{5}z^{-4})^{2} = (a^{5})^{2}(z^{-4})^{2}}$ On the left, we have ${a^{5}}$ to the exponent ${2}$ . Now ${5 \times 2 = 10}$ , so ${(a^{5})^{2} = a^{10}}$ Apply the ideas above to simplify the equation. $\dfrac{{(a^{5}z^{-4})^{2}}}{{(a^{-5}z^{2})^{-1}}} = \dfrac{{a^{10}z^{-8}}}{{a^{5}z^{-2}}}$ Break up the equation by variable and simplify. $\dfrac{{a^{10}z^{-8}}}{{a^{5}z^{-2}}} = \dfrac{{a^{10}}}{{a^{5}}} \cdot \dfrac{{z^{-8}}}{{z^{-2}}} = a^{{10} - {5}} \cdot z^{{-8} - {(-2)}} = a^{5}z^{-6}$